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Geometry of integrable systems with infinite degrees of freedom and new development of moduli theory.

Research Project

Project/Area Number 14102001
Research Category

Grant-in-Aid for Scientific Research (S)

Allocation TypeSingle-year Grants
Research Field Algebra
Research InstitutionKyoto University

Principal Investigator

UENO Kenji  Kyoto University, Graduate School of Science, Professor, 大学院理学研究科, 教授 (40011655)

Co-Investigator(Kenkyū-buntansha) KATO Fumiharu  Kyoto University, Graduate School of Science, Associated Professor, 大学院理学研究科, 助教授 (50294880)
KAWAGUCHI Shu  Kyoto University, Graduate School of Science, Assistant Professor, 大学院理学研究科, 助手 (20324600)
MOCHIZUKI Shinichi  Kyoto University, Research Institute for Mathematical Science, Professor, 数理解析研究所, 教授 (10243106)
TAKASAKI Kanehisa  Kyoto University, Graduate School of Human and Environmental Studies, Professor, 大学院人間環境学研究科, 教授 (40171433)
KATSURA Toshiyuki  University of Tokyo, Graduate School of Mathematical Science, Professor, 大学院数理科学研究科, 教授 (40108444)
木村 弘信  熊本大学, 大学院自然科学研究科, 教授 (40161575)
山田 泰彦  神戸大学, 理学部, 教授 (00202383)
江口 徹  東京大学, 大学院・理学系研究科, 教授 (20151970)
森脇 淳  京都大学, 大学院・理学研究科, 教授 (70191062)
加藤 和也  京都大学, 大学院・理学研究科, 教授 (90111450)
吉田 敬之  京都大学, 大学院・理学研究科, 教授 (40108973)
三輪 哲二  京都大学, 大学院・理学研究科, 教授 (10027386)
丸山 正樹  京都大学, 大学院・理学研究科, 教授 (50025459)
Project Period (FY) 2002 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥107,510,000 (Direct Cost: ¥82,700,000、Indirect Cost: ¥24,810,000)
Fiscal Year 2006: ¥19,500,000 (Direct Cost: ¥15,000,000、Indirect Cost: ¥4,500,000)
Fiscal Year 2005: ¥20,670,000 (Direct Cost: ¥15,900,000、Indirect Cost: ¥4,770,000)
Fiscal Year 2004: ¥20,800,000 (Direct Cost: ¥16,000,000、Indirect Cost: ¥4,800,000)
Fiscal Year 2003: ¥21,840,000 (Direct Cost: ¥16,800,000、Indirect Cost: ¥5,040,000)
Fiscal Year 2002: ¥24,700,000 (Direct Cost: ¥19,000,000、Indirect Cost: ¥5,700,000)
KeywordsIntegrable systems with infinite degrees of freedom / Moduli space / Conformal field theory / Modular functor / Topological field theory / Rigid geometry / Arithmetic geometry / Painleve equations / 政論幾何学 / 無限次元代数 / 代数曲線 / ヤコビ多様体 / 曲線の退化 / 重複ファイバー / K3曲面 / KZ方程式 / モノドロミー表現 / タイヒミュラー空間 / ワイル群 / 遠アーベル幾何学 / Rigid幾何学 / ミラー対称性 / モジュライ函手 / 位相場の理論 / カラビ・ヤウ多様体 / p進解析空間 / 幾何学的クリスタル / 3次元多様体の不変量 / リー代数 / Artin-Mazur形式群 / p進微分方程式
Research Abstract

From non-abelian conformal field theory (WSWN-model) and abelian conformal field Ueno and J.E. Andersen constructed modular functor and studied properties of the associated topological field theory. They also showed that S-matrices of non-abelian conformal field theory are determined by the genus 0 data. Moreover they showed that the Nielsen-Thurston classification of mapping class groups of 4-pointed spheres is determined by their quantum SU (n) representations. F. Kato and his collaborators have constructed the most general rigid geometry so that it can be applied to study moduli spaces by analytic method. S. Mochizuki has studied categorical aspect of moduli space of algebraic curves from different viewpoints. For example, he constructed theory of Frobenioids and that of etale theta functions, which give new direction of study of moduli spaces. Moreover, many interesting results on geometric and arithmetic properties of moduli spaces were obtained.
For theory of integrable systems K. Takasaki and his collaborators found that moduli spaces play important roles in the soliton theory. Also interesting relationship between geometry of special algebraic curves and Painleve equations were found. Moreover several important properties of quantum cohomology and quantum K-theory of flag manifolds have been found.
Our studies show that mathematical structure of moduli space is deeper than what we thought at the beginning and we need new mathematical tools for further investigations. Our results give a part of such tools and also show new directions to further investigations.

Report

(6 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • Research Products

    (46 results)

All 2007 2006 2005 2004 Other

All Journal Article (29 results) Book (3 results) Publications (14 results)

  • [Journal Article] Geometric construction of modular functors from conformal field theory2007

    • Author(s)
      J.E.Andersen, K.Ueno
    • Journal Title

      Journal of Knot theory and its Ramifications 16

      Pages: 127-202

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Geometric construction of modular functors from conformal field theory.2007

    • Author(s)
      J.E.Andersen, K.Ueno
    • Journal Title

      Journal of Knot theory and its Ramifications 16

      Pages: 127-202

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Geometric construction of modukar functor from conformal field theory2007

    • Author(s)
      J.E.Andersen, K.Ueno
    • Journal Title

      J. Knot Theory and its Ramification 16

      Pages: 127-202

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Topological quantum field theory and the Nielsen-Thurston classification of M(0,4)2006

    • Author(s)
      J.E.Andersen, G.Masbaum, K.Ueno
    • Journal Title

      Math. Proc. Camb. Phil. Soc. 141

      Pages: 477-488

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Rigid geometry and applications2006

    • Author(s)
      F.Kato, K.Fujiwara
    • Journal Title

      Advanced Studies in Pure Math. 45

      Pages: 325-384

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Topological quantum field theory and the Nielsen-Thurston classification of M(0,4).2006

    • Author(s)
      J.E.Andersen, G.Masbaum, K.Ueno
    • Journal Title

      Math. Proc. Camb. Phil. Soc. 141

      Pages: 477-488

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Topological quantum field theory and the Nielsen-Thurston classification of M(0,4)2006

    • Author(s)
      J.E.Andersen, G.Masbaum, K.Ueno
    • Journal Title

      Math. Proc. Cambridge Philos. Soc. 141

      Pages: 477-488

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Canonical heights, invariant currents, and dynamical eigensystems of morphisms for line bundles2006

    • Author(s)
      S.Kawaguchi
    • Journal Title

      J. Reine Angew. Math. 597

      Pages: 135-173

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Semi-graphs of Anabelioids2006

    • Author(s)
      S.Mochizuki
    • Journal Title

      Publ. Res. Inst. Math. Sc 42

      Pages: 221-322

    • NAID

      110004020852

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Explicit solutions of the classical Calogero& Sutherland systems for any root system2006

    • Author(s)
      R.Sasaki, K.Takasaki
    • Journal Title

      J. Math. Phys. 47-1

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On confluences of general hypergeometric systems2006

    • Author(s)
      H.Kimura, K.Takano
    • Journal Title

      Tohoku Math. J. 58-2

      Pages: 1-31

    • NAID

      110004308599

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Galois Sections in Absolute Anabelian Geometry2005

    • Author(s)
      S.Mochizuki
    • Journal Title

      Nagoya Math. J. 179

      Pages: 17-45

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Note on tautological classes of moduli of K3 surfaces2005

    • Author(s)
      G.van der Geer, T.Katsura
    • Journal Title

      Mosc. Math. J. 5

      Pages: 775-779

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Galois Sections in Absolute Anabelian Geometry.2005

    • Author(s)
      S.Mochizuki
    • Journal Title

      Nagoya Math.J. 179

      Pages: 17-45

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Note on tautological classes of moduli of K3 surfaces.2005

    • Author(s)
      G.van der Geer, T.Katsura
    • Journal Title

      Mosc.Math.J. 5

      Pages: 775-779

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Box-ball system with reflecting end.2005

    • Author(s)
      Yamada, Yasuhiko
    • Journal Title

      J.Nonlinear Math.Phys. 12-4

      Pages: 475-507

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Cubic pencils and Painleve Hamiltonians.2005

    • Author(s)
      Yamada, Yasuhiko
    • Journal Title

      Funkcial.Ekvac. 48-1

      Pages: 147-160

    • Related Report
      2005 Annual Research Report
  • [Journal Article] $q$-analogue of modified KP hierarchy and its quasi-classical limit2005

    • Author(s)
      Takasaki, Kanehisa
    • Journal Title

      Lett.Math.Phys. 72-3

      Pages: 165-181

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Elliptic spectral parameter and infinite-dimensional Grassmann variety.2005

    • Author(s)
      Takasaki, Kanehisa
    • Journal Title

      Progr.Math. 237

      Pages: 175-203

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Zur Entartung schwach verzweigter Gruppenoperationen auf Kurven.2005

    • Author(s)
      Kato, Fumiharu
    • Journal Title

      J.Reine Angew.Math. 589

      Pages: 201-236

    • NAID

      120006581747

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The $p$-adic icosahedron.2005

    • Author(s)
      Kato, Fumiharu
    • Journal Title

      Notices Amer.Math.Soc. 52-7

      Pages: 720-727

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Exterior differential algebras and flat connections on Weyl groups2005

    • Author(s)
      Anatol N.Kirillov, Toshiaki Maeno
    • Journal Title

      Proc.Japan Acad. 81,Ser.A

      Pages: 30-35

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Non-archimedean orbifolds covered by Mumford curves2005

    • Author(s)
      Fumiharu Kato
    • Journal Title

      Journal of Alg.Geom. 14

      Pages: 1-34

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Tyurin parameters and elliptic analogue of nonlinear Schrodinger hierarchy2004

    • Author(s)
      K.Takasaki
    • Journal Title

      J. Math. Sci. Univ. Tokyo 11

      Pages: 91-131

    • NAID

      110000974890

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Tyurin parameters and elliptic analogue of nonlinear Schrodinger hierarchy.2004

    • Author(s)
      K.TakasakI
    • Journal Title

      J.Math.Sci.Univ.Tokyo 11

      Pages: 91-131

    • NAID

      110000974890

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] p-adic Hodge theory and values of zeta functions of modular forms. Cohomologies p-adiques et applications arithmetiques. III2004

    • Author(s)
      Kazuya Kato
    • Journal Title

      Asterisque 295

      Pages: 117-290

    • Related Report
      2004 Annual Research Report
  • [Journal Article] The Geometry of Anabelioids2004

    • Author(s)
      Shinichi Mochizuki
    • Journal Title

      Publ.Res.Inst.Math.Sci. 40

      Pages: 819-881

    • NAID

      110001050927

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Constructing modular functors from conformal field theories

    • Author(s)
      Jorgen Ellegaard Andersen, Kenji Ueno
    • Journal Title

      J.of Knot theory and it's ramifications (近刊予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Topological Quantum Field Theory and the Nielsen-Thurston classification of M(0,4)

    • Author(s)
      Jorgen Ellegaard Andersen, Gregor Masbaum, Kenji Ueno
    • Journal Title

      Geometry and Topology (近刊予定)

    • Related Report
      2004 Annual Research Report
  • [Book] 共形場理論入門2006

    • Author(s)
      山田 泰彦
    • Total Pages
      265
    • Publisher
      培風館
    • Related Report
      2005 Annual Research Report
  • [Book] 代数幾何学2005

    • Author(s)
      上野 健爾
    • Total Pages
      626
    • Publisher
      岩波書店
    • Related Report
      2005 Annual Research Report
  • [Book] ツイスターの世界2005

    • Author(s)
      高崎 金久
    • Total Pages
      266
    • Publisher
      共立出版
    • Related Report
      2005 Annual Research Report
  • [Publications] Cornelissen, Gunther, Kato, Fumiharu: "Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic"Duke Math.J.. 116. 431-470 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takesaki, Kanehisa: "Spectral curve, Darboux coordinates and Hamiltonian structure of periodic dressing chains"Commun.Math.Phys.. 241. 111-142 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Miwa, Tetsuji others: "Particle content of the (k,3)-configurations"Publ.Res.Inst.. 40. 163-220 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Ito, Yukari: "The McKay correspondence---a bridge from algebra to geometry"European women in mathematics(Malta). 127-147 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Yamada, Y. othres: "Geometric crystal and tropical R for D^<(1)>_n"Int.Math.Res.Not.. 48. 2565-2620 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Mochizuki, Shinichi: "Topics surrounding the anabelian geometry of hyperbolic curves. Galois groups and fundamental groups"Math.Sci.Res.Inst.Publ.. 41. 119-165 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Ueno, Kenji: "Algebraic geometry.3.Further study of schemes"AMS. 222 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Miwa Tetsuji, others: "Higher normal functions and Griffiths groups"Compositio Math.. 134,No.2. 193-241 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Moriwaki, Atsushi: "Nef divisors in codimension one on the moduli space curves"Compositio Math.. 132,No.2. 191-228 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Mochizuki, Shinichi: "A survery of the Hodge-Arakelove theory of elliptic curves"Proc.Sympos.Pure Math.. 70. 533-569 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Takasaki, K., others: "Hierarchy of (2+1)-dimensional nonlinear shrodinger equation, self-dual Yang-Mills equation, and toroidal Lie algebras"Ann.Henri Poincare. 3,no.5. 817-845 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Yamada, Yasuhiko, others: "Backlund transformations and the manifolds of Painleve systems"Funkcial.Ekvac.. 45,no2. 237-258 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Ito, Yukari: "Special McKay correspondence, Geometry of Toric Varietie (Grenoble 2000)"Seminaires et Congres. 6. 213-223 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Ueno, Y.Shimizu: "Advances in Moduli Theory"AMS. 300 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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